The gradient flow of Higgs pairs

Abstract

In this paper, we consider the gradient flow of the Yang-Mills-Higgs functional of Higgs pairs on a Hermitian vector bundle $(E , H\_{0})$ over a Kähler surface $(M , \omega )$, and study the asymptotic behavior of the heat flow for Higgs pairs at infinity. The main result is that the gradient flow with initial condition $(A\_{0} , \phi\_{0})$ converges, in an appropriate sense which takes into account bubbling phenomena, to a critical points $(A\_{\infty } , \phi\_{\infty})$ of this functional. We also prove that the limiting Higgs pair $(A\_{\infty }, \phi\_{\infty})$ can be extended smoothly to a vector bundle $E\_{\infty }$ over $(M , \omega )$ , and the isomorphism class of the limiting Higgs bundle $(E\_{\infty } , A\_{\infty } , \phi\_{\infty})$ is given by the double dual of the graded Higgs sheaves associate to Harder-Narasimhan-Seshadri filtration of the initial Higgs bundle $(E , A\_{0} , \phi\_{0})$.